\(\mathcal {T}_g\)  has the structure of a complex manifold which is homeomorphic to an open ball in \({3g-3}\) . Traditional approach to constructing the complex structure is by Bers embedding of \(\mathcal {T}_g\)  into the complex vector space of holomorphic quadratic differentials on a (reference) compact Riemann surface of genus \(g\) , see chapter 6 of [1]} for details. A differential geometric approach to the complex structure is given by the Weil-Petersson metric, see section 3 and theorem 3.3 in [2]}. In the algebro-geometric approach one shows that for all \([(X, f)] \in \mathcal {T}_g\)  there exists a neighbourhood which can be identified with the base space of a Kuranishi family of (equivalence classes of) marked compact Riemann surfaces of genus \(g\)  with central fiber isomorphic to \([(X, f)]\) . By the universal property of these Kuranishi families, and the fact that a marked Riemann surface has trivial automorphisms, these local neighborhoods can be glued together into a single Kuranishi family, thus giving \(\mathcal {T}_g\)  the structure of a complex manifold, see [3]} chapter XV for details. In all the three topologies induced by these three complex structures, it can be shown that \(\mathcal {T}_g\)  is an open ball in \({3g-3}\) . From now on we consider \(\mathcal {T}_g\)  as a complex manifold with complex structure induced from the Bers embedding.
